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Full text of "Men Of Mathematics"

mil be few instances where we shall fail to arrive at propositions
of more or less importance, even when the complication of the
calculations precludes a complete answer to the problem.'
He goes on to say that this, the true scientific method to be
followed, has been but little used owing to the extreme compli*
cation of the calculations (algebraic) which it entails; 4but', he
adds, In many instances this complication is only apparent and
vanishes after the first attack.' He continues:
ftl have treated several branches of analysis in this manner,
and although I have often set myself problems beyond my
powers, I have nevertheless arrived at a large number of general
results which throw a strong light on the nature of those quan-
tities whose elucidation is the object of mathematics. On
another occasion I shall give the results at which I have
arrived in these researches and the procedure which has led me
to them. In the present memoir I shall treat the problem of the
algebraic solution of equations in all its generality.*
Presently he states two general inter-related problems which
he proposes to discuss:
*1. To find all the equations of any given degree which are
solvable algebraically.
2. To determine whether a given equation is or is not solv-
able algebraically.'
At bottom, he says, these two problems are the same, and
although he does not claim a complete solution, he does indicate
an. infallible method (des moyens stirs) for disposing of them
Abel's irrepressible inventiveness hurried him on to vaster
problems before he had time to return to these; their complete
solution - the explicit statement of necessary and sufficient
conditions that an algebraic equation be solvable algebraically
- was to be reserved for Galois. When this memoir of Abel's was
published in 1828, Galois was a boy of sixteen, already well
started on his career of fundamental discovery. Galois later
came to know and admire the work of Abel; it is probable that
Abel never heard the name of Galois, although when Abel
visited Paris he and his brilliant successor could have been only
a few miles apart. But for the stupidity of Galois' teachers and